Digoxin is a drug used to treat heart disease. Doctors must prescribe an amount of medication that will keep the concentration of digoxin in the blood above effective levels without exceeding safe levels. The following table shows the amount of digoxin remaining in the blood of a given patient at various times ( days ) after an initial dose ( mg ) of digoxin.

　　(1) Try to construct an approximate functional relationship between the digoxin content in the blood and the number of days after taking the drug;

　　(2) Predict the digoxin content in the blood days later.

　　Solution：(1) Make a scatter plot based on the given data (see the scatter points in the figure below) .

[ Experimental Detailed Explanation ]

 [ Scatter Plot and Linear Regression ]

　　It can be seen from this figure that there is roughly an exponential functional relationship between and , so the functional relationship expression is

,

Among them ,  and  are undetermined constants. Taking the logarithm on both sides of the above equation , we get

,

Let , , then the exponential function is converted into a linear function

.

Using the title data table to further calculate the following table:

　　Calculating using a method similar to the above example, we get

, .

so

.

　　Then from the relation , we get

,

Thus, the approximate functional relationship between the digoxin content in the blood and the number of days after taking the drug is obtained:

.

Adding the above regression curve to the scatter plot , it can be seen that the exponential function fits the scatter plot quite well, indicating that the exponential model is reasonable.

　　(2) According to the above functional relationship, the digoxin content in the blood after days is

( mg ) .